Problem 34 - Elementary Algebra

Question

For the following problems, classify each polynomial as a monomial, binomial, or trinomial. State the degree of each polynomial and write the numerical coefficient of each term. $$ 4 x y^{3} z^{5} w^{0}, \quad w \neq 0 $$

Step-by-Step Solution

Answer

Question: Classify the polynomial \(4xy^3z^5w^0\), state its degree, and write the numerical coefficient of each term, with \(w\neq0\). Answer: The polynomial is a monomial with a degree of 9 and a numerical coefficient of 4.

Step 1: Rewriting the expression

Since any variable raised to the power of 0 is equal to 1, we can simplify the expression by removing the \(w^0\) term. The expression becomes: $$ 4xy^3z^5 $$

Step 2: Classify the Polynomial

The given expression is a single term with variables \(x\), \(y\), and \(z\). Since there is only one term, the expression is classified as a monomial.

Step 3: Determine the Degree

To find the degree of the monomial, we sum the exponents of all the variables involved. In this case, the degree of the monomial is \(1+3+5 = 9\).

Step 4: Identify the Numerical Coefficient

The numerical coefficient of the monomial is the constant value in front of the variables. In this case, the numerical coefficient is 4.

Step 5: Summary

The given polynomial is a monomial with a degree of 9. The numerical coefficient of the monomial is 4.

Key Concepts

Monomials

A monomial is a type of polynomial that consists of just one term. In a mathematical expression, if you see that there’s only one grouping of numbers and variables, it's a monomial.
Consider the expression \(4xy^3z^5\). This is a classic example of a monomial.

• Monomials can have multiple variables, like \(x\), \(y\), and \(z\), and these variables can be raised to different powers.
• The key point is that these elements are all part of a single term.

Since there’s only one term, this expression cannot be considered a binomial or trinomial, as those would have two or three terms, respectively.
It's important to recognize monomials so that you can apply certain rules and techniques specific to them in algebra.

Degree of a Polynomial

The degree of a polynomial refers to the highest sum of the exponents of the variables in a term. This is a crucial concept because the degree can tell you a lot about the behavior and characteristics of the polynomial at large values.
In our example, the expression \(4xy^3z^5\) is a monomial. Here’s how you find its degree:

• Look at the exponents of each variable: \(x^1\), \(y^3\), and \(z^5\).
• Add these exponents together: \(1 + 3 + 5 = 9\).

Thus, the degree of this monomial is 9.
Always remember that the degree can influence the shape of the graph of the polynomial and how it behaves for larger values of the variable involved.

Numerical Coefficients

The term "numerical coefficient" refers to the numerical part of a polynomial, which multiplies the variables.
In simpler words, it’s the number in front of the variables or the constant factor in a term.
For our example, in the expression \(4xy^3z^5\), the numerical coefficient is 4. It’s that easy:

• No matter how complex a polynomial term looks, the numerical coefficient is the easy-to-spot number that precedes the variables.
• This coefficient allows you to understand how "strongly" the variables are being affected when calculating the polynomial's value.

Numerical coefficients play a vital role in operations involving polynomials, such as addition, subtraction, or scalar multiplication, because they allow us to easily combine like terms or factor expressions.